Lesser Moonstone can be obtained from either prospecting Tin Ore or Iron Ore. Either way, you'll be competing with obtaining it over multiple other gems as well. Good news is, also either way, you're still going to net a decent amount of these, given you have a lot of either ore.
Malachite is one of the first gems you'll be able to get in the game, and you'll be needing them in order to level up your Jewelcrafting, or the other professions noted.
Shadowgem is the main gem you're going to get from prospecting Tin Ore. I've noticed it can go either way with being a nice sale on the auction house versus nice sales on the crafted equipment it can create, but know that it doesn't have as many good items for crafting.
Tigerseye is one of the first gems you'll gather, and seeing as it's an easier find when prospecting Copper Ore, there's a greater chance it'll be cheap on the auction house.